American banking chiefs are facing the biggest, most nail-biting assessment of their lives: a series of financial "stress tests" by teams of government experts to see whether their institutions can be salvaged. Saving the banking system and boosting the economy is the first big challenge for Barack Obama's new administration.

Forensic inspectors from at least four federal agencies will soon begin combing through asset registers, trading ledgers and balance sheets at the 18 or so biggest US banks. Line by line, the inspectors will tot up billions of dollars in liabilities.

As Wall Street's institutions perform a seemingly endless dance on the brink of insolvency, the Obama administration is under increasing pressure to consider the unthinkable. Frustrated policymakers, financial experts and voters are suggesting it is time to cut off life support to troubled banks.

"In a lot of cases, they're clearly bankrupt," said Dean Baker, co-director of the Centre for Economic and Policy Research. "Citigroup and Bank of America have a cesspool of bad assets and without government support, they probably would have failed. You have to ask yourself what we're trying to do here."

Wall Street's broken banks present policymakers with a series of difficult challenges that are echoed around the world.

The White House is trying to limit damage. Obama's spokesman, ­Robert Gibbs, assured reporters: "This plan wasn't written in one day and ­neither should it be judged by one day's stock movements."

But few experts are confident the strategy will work, or even that ­Geithner's sketchy outline amounts to a viable proposal. Paul Miller, a banking analyst at FBR Capital Markets, described it as "nowhere near enough" with too many gaps in the detail: "It lacks sufficient transparency to bring confidence and private equity back to financials."

At the heart of Geithner's proposal is the creation of a so-called "bad bank" of up to $1tn. The treasury will seed this with taxpayers' money in the hope of luring private funds to purchase the banks' toxic assets. Geithner says he has yet to determine quite how this will be structured. Under fire, he admitted to the Senate banking committee: "I understand the desire for details and I understand the disappointment about the lack of details."

Heavyweight critics

Only two weeks after being sworn in, Geithner has attracted a roster of heavyweight critics. The Nobel prize-winning economist Paul Krugman and the influential harbinger of the credit crunch, ­Nouriel Roubini, have called for large-scale nationalisation of institutions rather than ­further efforts to keep them afloat through infusions of taxpayers' money.

"The problem is not toxic assets," said Krugman. "The problem is that financial institutions have lost a lot of money and many of the big ones, if they are not actually insolvent, are very close."

Aware of the mounting frustration and impatience, banking chiefs are trying to be humble. The head of Bank of America, Ken Lewis, opted for an eight-hour train journey from his North Carolina headquarters to appear before Congress on Wednesday, anxious to avoid using his corporate jet.

But as Lewis and seven other bank bosses arrived on Capitol Hill, they ran the gauntlet of protesters from Jesse Jackson's Rainbow-Push coalition, who were railing against the treasury's seemingly unfettered support for Wall Street.

Morgan Stanley's boss, John Mack, had a taste of public anger when campaigners descended on his mansion in Rye, a wealthy enclave in New York's commuter belt. Waving placards, the early-morning demonstrators accused him of being a "loan shark". "You've got to hold these chief executives personally responsible," said Bruce Marks, chief executive of the Neighbourhood Assistance Corporation of America, which organised the protest.

The conventional wisdom has been that the failure of top banks would provoke a catastrophic chain of events similar to the fallout from Lehman Brothers' collapse: a stockmarket crash, a panicked withdrawal of funds and a reverberation in liabilities around intertwined institutions in the US and abroad that could send the financial system into meltdown.

By trying to cleanse banks' balance sheets, the US government has been hoping to restore top financial firms to health. But in order to sell toxic assets such as mortgage-related securities, banks would first have to record their true value. Many are still held on balance sheets at vastly inflated prices.

A source at one Wall Street firm said Geithner's plan would either involve buying assets at over-egged values or would force banks to take billions of dollars in life-threatening write-downs: "The real question is are these treasury guys really serious about forcing the commercial banks to mark down their portfolios?"

By ordering so-called stress tests of each firm, the treasury intends to determine the extent of hidden losses. Geithner has yet to say what he will do with institutions that fail their tests – but a form of nationalisation, followed by dismemberment and resale, is one option.

Douglas Elliott, a banking expert at the Brookings Institution in Washington, said this would be risky: "If the reason for stepping in is because these banks are in a deep black hole, that black hole would become explicitly the taxpayers' black hole."

Nationalisation would wipe out shareholders – and Elliott said that as soon as one firm was taken out, others would suffer a crisis of confidence: "As soon as you do this to the weakest banks, the next weakest banks' stock will start to fall heavily. People will think 'maybe it's just a week or two before the government does this to us'." Analysts at Goldman Sachs have estimated banks could lose $2.1tn on delinquent mortgages, credit cards and loans before the crisis ends. Confidence is at rock bottom.

"The problem is that none of us – no Americans – trust you any more," the Democratic congressman Michael Capuano told Wall Street bosses this week, declaring that he would not be depositing a cent in any of the top eight banks. "I don't want my money put into CDOs, credit default swaps and humongous bonuses."